Question: $ A = \left[\begin{array}{rr}2 & -2 \\ 5 & 4\end{array}\right]$ $ F = \left[\begin{array}{rrr}3 & 2 & -1 \\ 1 & 2 & 3\end{array}\right]$ What is $ A F$ ?
Answer: Because $ A$ has dimensions $(2\times2)$ and $ F$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ A F = \left[\begin{array}{rr}{2} & {-2} \\ {5} & {4}\end{array}\right] \left[\begin{array}{rrr}{3} & \color{#DF0030}{2} & \color{#9D38BD}{-1} \\ {1} & \color{#DF0030}{2} & \color{#9D38BD}{3}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{2}\cdot{3}+{-2}\cdot{1} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rrr}{2}\cdot{3}+{-2}\cdot{1} & ? & ? \\ {5}\cdot{3}+{4}\cdot{1} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rrr}{2}\cdot{3}+{-2}\cdot{1} & {2}\cdot\color{#DF0030}{2}+{-2}\cdot\color{#DF0030}{2} & ? \\ {5}\cdot{3}+{4}\cdot{1} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{2}\cdot{3}+{-2}\cdot{1} & {2}\cdot\color{#DF0030}{2}+{-2}\cdot\color{#DF0030}{2} & {2}\cdot\color{#9D38BD}{-1}+{-2}\cdot\color{#9D38BD}{3} \\ {5}\cdot{3}+{4}\cdot{1} & {5}\cdot\color{#DF0030}{2}+{4}\cdot\color{#DF0030}{2} & {5}\cdot\color{#9D38BD}{-1}+{4}\cdot\color{#9D38BD}{3}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}4 & 0 & -8 \\ 19 & 18 & 7\end{array}\right] $